{
  "id": "math/0603273",
  "version": "v2",
  "published": "2006-03-12T03:13:44.000Z",
  "updated": "2006-06-29T14:32:21.000Z",
  "title": "Governing Singularities of Schubert Varieties",
  "authors": [
    "Alexander Woo",
    "Alexander Yong"
  ],
  "comment": "23 pages. Software available at the authors' webpages. Version 2 is the submitted version. It has a nomenclature change: \"Bruhat-restricted pattern avoidance\" is renamed \"interval pattern avoidance\"; the introduction has been reorganized",
  "journal": "J. Algebra, Vol. 320 (2008), No. 2, p. 495--520",
  "doi": "10.1016/j.jalgebra.2007.12.016",
  "categories": [
    "math.AG",
    "math.AC",
    "math.CO"
  ],
  "abstract": "We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of *interval pattern avoidance*. For \"reasonable\" invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P=\"singular\"; classical pattern avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [Woo-Yong'05]; the description of the singular locus (which was independently proved by [Billey-Warrington '03], [Cortez '03], [Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with *Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-06-29T14:32:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M15",
      "14M05",
      "05E99"
    ],
    "keywords": [
      "schubert variety",
      "governing singularities",
      "pattern avoidance applies admirably",
      "computational commutative algebra methodology",
      "classical pattern avoidance applies"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......3273W"
    }
  }
}